In the 1923 seminal paper by Burchnall and Chaundy [1]}, the authors proposed to describe all pairs of commuting differential operators that are not simply contained in a polynomial ring \({\bf C}[M]\)Although the field of coefficients is not mentioned in [1]}, we work over the complex numbers \({\bf C}\)  in this paper unless
otherwise specified., \(M\in \mathcal {D}\)  (cf. Section  below for
notation). We note that, whenever two differential operators \(A\) , \(B\) , commute
with an operator \(L\)  of order greater than zero, then they commute with each
other (cf. Corollary REF ), and therefore
maximal-commutative subalgebras of \(\mathcal {D}\)  are centralizers; these are the main
objects we seek to classify. In addition,
we will always assume that a commutative subalgebra
contains a normalized element
\(L=\partial ^n+u_{n-2}\partial ^{n-2}+\cdots +u_0\) , although
some proviso is needed (cf., e.g., [3]}), except in the `formal' case
when the coefficients are
just taken to be formal power series.
We will say that the Burchnall–Chaundy \((\) BC\()\)  problem asks when the
centralizer \({\mathcal {C}}_{\mathcal {D}}(L)\)  of an operator \(L\)  is not a polynomial ring
(which we regard as a `trivial' case, for example \({\bf C}[G]\) ,
with \(L\)  a power of some \(G\in \mathcal {D}\) ) and we call such an \(L\) 
a “BC solution”. Burchnall and Chaundy immediately make the observation that if the orders of
two commuting \(L\)  and \(B\)  are coprime, then either one is a BC solution.
Eventually [4]}, they were able to classify the commutative subalgebras
\({\bf C}[L,B]\)  of rank one – the rank, defined in
Section REF  for any subset of \(\mathcal {D}\) , is the greatest common
divisor of the orders of all elements of \({\bf C}[L,B]\) .
The classification problem is wide open in higher (than one) rank, although a theoretical geometric description was given [5]}, [6]}.
